\documentclass[UTF8,10pt]{ctexart}

\usepackage{amsmath}
\usepackage{bm}

\begin{document}

向量点乘后对向量求导

\begin{equation}
    \frac{\partial (\bm{a} \cdot \bm{b})}{\partial \bm{c}} = \left( \frac{\partial \bm{a}}{\partial \bm{c}} \right)^{\mathrm{T}} \bm{b} + \left( \frac{\partial \bm{b}}{\partial \bm{c}} \right)^{\mathrm{T}} \bm{a}
\end{equation}

\begin{equation}
    \frac{\partial (u \bm{a})}{\partial \bm{b}} = \bm{a} \left( \frac{\partial u}{\partial \bm{b}} \right)^{\mathrm{T}} + u \left( \frac{\partial \bm{a}}{\partial \bm{b}} \right)
\end{equation}

\begin{equation}
    \frac{\partial \bm{a}}{\partial \bm{a}} = \bm{I}
\end{equation}

向量叉乘后对向量求导，这个问题较为复杂，目前来看应该是这样的
\begin{equation}
    \frac{\partial (\bm{a} \times \bm{b})}{\partial \bm{c}} = \underline{\widetilde{\bm{a}}} \frac{\partial \bm{b}}{\partial \bm{c}} - \underline{\widetilde{\bm{b}}} \frac{\partial \bm{a}}{\partial \bm{c}}
\end{equation}
其中
\begin{equation}
    \underline{\widetilde{\bm{a}}} = \left[
        \begin{array}{ccc}
            0 & -a_3 & a_2 \\
            a_3 & 0 & -a_1 \\
            -a_2 & a_1 & 0
        \end{array}
    \right]
\end{equation}

\end{document}

